One-Sample Hypothesis Tests Chapter99 Logic of Hypothesis Testing Statistical Hypothesis Testing Testing a Mean: Known Population Variance Testing a Mean: Unknown Population Variance Testing a Proportion Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin

Logic of Hypothesis Testing Steps in Hypothesis TestingSteps in Hypothesis Testing Step 1: State the assumption to be tested Step 2: Specify the decision rule Step 3: Collect the data to test the hypothesis Step 4: Make a decision Step 5: Take action based on the decision 9-2

Logic of Hypothesis Testing Hypotheses are a pair of mutually exclusive, collectively exhaustive statements about the world.Hypotheses are a pair of mutually exclusive, collectively exhaustive statements about the world. H 0 : Null Hypothesis H 1 : Alternative HypothesisH 0 : Null Hypothesis H 1 : Alternative Hypothesis Efforts will be made to reject the null hypothesis. The null hypothesis is assumed true and a contradiction is sought.Efforts will be made to reject the null hypothesis. The null hypothesis is assumed true and a contradiction is sought. If H 0 is rejected, we tentatively conclude H 1 to be the case.If H 0 is rejected, we tentatively conclude H 1 to be the case. State the Hypothesis State the Hypothesis 9-3

Logic of Hypothesis Testing Types of Error Types of Error Type I error: Rejecting the null hypothesis when it is true. This occurs with probability .Type I error: Rejecting the null hypothesis when it is true. This occurs with probability . Type II error: Failure to reject the null hypothesis when it is false. This occurs with probability .Type II error: Failure to reject the null hypothesis when it is false. This occurs with probability . 9-4

Statistical Hypothesis Testing A statistical hypothesis is a statement about the value of a population parameter .A statistical hypothesis is a statement about the value of a population parameter . A hypothesis test is a decision between two competing mutually exclusive and collectively exhaustive hypotheses about the value of .A hypothesis test is a decision between two competing mutually exclusive and collectively exhaustive hypotheses about the value of . Left-Tailed Test Right-Tailed Test Two-Tailed Test 9-5

Statistical Hypothesis Testing The direction of the test is indicated by H 1 :The direction of the test is indicated by H 1 : > indicates a right-tailed test < indicates a left-tailed test ≠ indicates a two-tailed test 9-6

Statistical Hypothesis Testing Decision Rule Decision Rule A test statistic shows how far the sample estimate is from its expected value, in terms of its own standard error.A test statistic shows how far the sample estimate is from its expected value, in terms of its own standard error. The decision rule uses the known sampling distribution of the test statistic to establish the critical value that divides the sampling distribution into two regions.The decision rule uses the known sampling distribution of the test statistic to establish the critical value that divides the sampling distribution into two regions. Reject H 0 if the test statistic lies in the rejection region.Reject H 0 if the test statistic lies in the rejection region. 9-7

Statistical Hypothesis Testing Decision Rule for Two-Tailed Test Decision Rule for Two-Tailed Test Reject H 0 if the test statistic right-tail critical value.Reject H 0 if the test statistic right-tail critical value. Figure Critical value+ Critical value 9-8

Statistical Hypothesis Testing Decision Rule for Left-Tailed Test Decision Rule for Left-Tailed Test Reject H 0 if the test statistic < left-tail critical value.Reject H 0 if the test statistic < left-tail critical value. Figure Critical value 9-9

Statistical Hypothesis Testing Decision Rule for Right-Tailed Test Decision Rule for Right-Tailed Test Reject H 0 if the test statistic > right-tail critical value.Reject H 0 if the test statistic > right-tail critical value. + Critical value Figure

Testing a Mean: Known Population Variance The test statistic compares the sample mean x with the hypothesized mean  0.The test statistic compares the sample mean x with the hypothesized mean  0. The difference between x and  0 is divided by the standard error of the mean (denoted  x ).The difference between x and  0 is divided by the standard error of the mean (denoted  x ). The test statistic isThe test statistic is z =z =z =z = x –  0  / n 9-11

Testing the Hypothesis Testing the Hypothesis Make the decision: If the test statistic falls in the rejection region as defined by the critical value, we reject H 0 and conclude H 1.Make the decision: If the test statistic falls in the rejection region as defined by the critical value, we reject H 0 and conclude H 1. Using the p-Value Approach Using the p-Value Approach The p-value is the probability of the sample result (or one more extreme) assuming that H 0 is true. Using the p-value, we reject H 0 if p-value < . The p-value is the probability of the sample result (or one more extreme) assuming that H 0 is true. Using the p-value, we reject H 0 if p-value < . Testing a Mean: Known Population Variance 9-12

Testing a Mean: Unknown Population Variance When the population standard deviation  is unknown and the population may be assumed normal, the test statistic follows the Student’s t distribution with = n – 1 degrees of freedom.When the population standard deviation  is unknown and the population may be assumed normal, the test statistic follows the Student’s t distribution with = n – 1 degrees of freedom. The test statistic isThe test statistic is t calc = x –  0 s/ n s/ n Using Student’s t Using Student’s t 9-13

Testing a Proportion p =p =p =p =xn= number of successes sample size z calc = p –  0  p  p If n  0 > 10 and n(1-  0 ) > 10, then the test statistic isIf n  0 > 10 and n(1-  0 ) > 10, then the test statistic is Where  p =  0 (1-  0 ) n z calc z calc will be compared to a critical value depending on . 9-14

Testing a Proportion Reject the null hypothesis if the test statistic falls in the rejection region.Reject the null hypothesis if the test statistic falls in the rejection region. Using the p-value, we reject H 0 if p-value < . Using the p-value, we reject H 0 if p-value < . NOTE: 1. A two-tailed hypothesis test at the 5% level of significance (  =.05) is exactly equivalent to asking whether the 95% confidence interval for the mean (proportion) includes the hypothesized mean (proportion). 2. If the confidence interval includes the hypothesized mean (proportion), then we cannot reject the null hypothesis. 9-15